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A strange functional equation

Source: Korea junior mathematical olympiad 2nd round 2019 december problem 6

November 17, 2019
algebrafunctional equationKJMOfunction

Problem Statement

Find all functions f:Rā†’Rf:\mathbb{R} \rightarrow \mathbb{R} which satisfies the followings. (Note that R\mathbb{R} stands for the set of all real numbers) (1) For each real numbers xx, yy, the equality f(x+f(x)+xy)=2f(x)+xf(y)f(x+f(x)+xy) = 2f(x)+xf(y) holds. (2) For every real number zz, there exists xx such that f(x)=zf(x) = z.
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